n Wordbank n Chapter outline NEW CENTURY MATHS ADVANCED for the Australian Curriculum9

Transcription

1 3Number nd lgebr Produts nd ftors Algebr is the brnh of mthemtis tht uses symbols nd formuls to desribe number ptterns nd reltionships in our nturl nd physil world. In 825 CE, the Persin mthemtiin l-khwrizmi used the Arbi word l-jbr to desribe the proess of dding equl quntities to both sides of n eqution. When l-khwrizmi s book ws trnslted into Ltin nd introdued to Europe, l-jbr beme lgebr nd the word ws dopted s the nme for the whole subjet.

2 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Shutterstok.om/mxms n Chpter outline Profiieny strnds 3-01 Adding nd subtrting terms U F R C 3-02 Multiplying nd dividing terms U F R C 3-03 Adding nd subtrting lgebri frtions U F R C 3-04 Multiplying nd dividing lgebri frtions U F R C 3-05 Expnding expressions U F R C 3-06 Ftorising expressions U F R C 3-07 Expnding binomil produts U F R C 3-08 Perfet squres* U F R C 3-09 Differene of two squres* U F R C 3-10 Mixed expnsions* F R C 3-11 Ftorising speil binomil produts* U F R C 3-12 Ftorising qudrti expressions* U F R C 3-13 Ftorising qudrti expressions of the form x 2 þ bx þ * U F R C 3-14 Mixed ftoristions* F R C 3-15 Ftorising lgebri frtions* U F R C n Wordbnk binomil An lgebri expression tht onsists of two terms, for exmple, 4 þ 9, 3 y, x 2 4x ftorise To rewrite n expression with grouping symbols, by tking out the highest ommon ftor; ftorising is the opposite of expnding; for exmple, 9r 2 þ 36r ftorised is 9r(r þ 4) highest ommon ftor (HCF) The lrgest term tht is ftor of two or more terms, for exmple, the HCF of 9r 2 þ 36r is 9r perfet squre A squre number or n lgebri expression tht represents one, for exmple, 64, (x þ 9) 2 qudrti expression An lgebri expression in whih the highest power of the vrible is 2, for exmple, 2x 2 þ 5x 3orx 2 þ 2 *STAGE 5.3

3 Chpter Produts nd ftors n In this hpter you will: simplify lgebri expressions involving the four opertions pply the distributive lw to the expnsion of lgebri expressions, inluding binomils, nd ollet like terms where pproprite dd, subtrt, multiply nd divide simple lgebri frtions ftorise lgebri expressions (STAGE 5.3) expnd binomil produts (STAGE 5.3) ftorise qudrti expressions (STAGE 5.3) ftorise nd simplify expressions involving lgebri frtions SkillChek Worksheet StrtUp ssignment 3 MAT09NAWK10025 Puzzle sheet Generlised rithmeti MAT09NAPS10026 Skillsheet Algebri expressions MAT09NASS10012 Puzzle sheet Substitution puzzle MAT09NAPS Write n lgebri expression for eh sttement. The ost of p books t $Q eh. b The selling prie fter GST of $t is dded to the mrked prie of $40. The number of metres in h km. d The verge of m, p, v nd w. e The number of dollrs in y ents. 2 If f ¼ 5, g ¼ 4nd r ¼ 2, then evlute eh expression. 2f þ g b 8 þ rf f 2 r 2 d 3g 2 5 e r(f g) f 5r þ f 3 3 If the perimeter of retngle with length l nd width w hs the formul P ¼ 2l þ 2w, find the perimeter of retngle with length 4.7 m nd width 2.5 m. 4 Simplify eh expression. 5n þ 6n b 9h 3h 10pq qp d x 2 þ 4x 2 e 4 3 w f 8 3 5x g 2g 3 3h h 3y 3 6y i 10x 4 2 j 6b 4 b k 16xy 8x 5 Find the highest ommon ftor (HCF) of eh pir of numbers. 8 nd 12 b 18 nd nd 30 6 Evlute eh expression b l 2n 2 4n d 3 8 þ

4 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum Adding nd subtrting terms Summry Skillsheet Algebr using digrms MAT09NASS10013 Only like terms n be dded or subtrted. For exmple, 2x 5y nnot be simplified beuse x nd y represent different numbers. Exmple 1 Simplify eh expression. 5r þ 4s þ 7r 2s b 8u 2 4u þ 3u 7u 2 b 5r þ 4s þ 7r 2s ¼ 5r þ 7r þ 4s 2s ¼ 12r þ 2s Group the like terms. 12r nd 2s nnot be simplified further. 8u 2 4u þ 3u 7u 2 ¼ 8u 2 7u 2 4u þ 3u Eh þ nd sign belongs to ¼ u 2 u the term tht follows them. Exmple 2 Write simplified lgebri expression for the perimeter of this retngle. 2x Perimeter ¼ x þ 3 þ 2x þ x þ 3 þ 2x ¼ x þ 2x þ x þ 2x þ 3 þ 3 ¼ 6x þ 6 x + 3 Adding the lengths of the 4 sides. Group the like terms. Exerise 3-01 Adding nd subtrting terms 1 Simplify eh lgebri expression. 2fg þ 3fg 4fg b 6uvw uvw þ 4uvw 3t þ 3 þ 5t d 3x þ 2y þ 8y e 6g þ h 4g f 2fg þ 3f 4fg g 7p 2 3 2p 2 h 4p q 3q i 8r þ 6r þ 3 j 3 þ 4b 7 k 4v þ 3 þ 2v þ 7 l 2y þ 6x þ y 3x m 7p þ 8q p q n 8 3w þ 7 2w o 2n 3 12n þ 5 p 8e 3f 2e 4f q 2l 2 þ 2l þ 3l 2 6l r 9b 12b 2 þ 6b 2 15b 2 Simplify 15 2 þ 12 þ Selet the orret nswer A, B, C or D. A 24 þ 48 B 27 2 þ 30 C þ 48 D 15 2 þ See Exmple 1 69

5 Chpter Produts nd ftors See Exmple 2 3 Use the substitutions ¼ 2, b ¼ 3 nd ¼ 4 to test whether eh eqution is orret or inorret ¼ 8 b 9b þ b ¼ 9b 2 7 þ 3b ¼ 10 þ þ b d 3 2 þ 5 2 ¼ 8 2 e 4b þ 3b ¼ 7b f 12 ¼ 12 g 3 5 ¼ 2 h 5b þ 2b 2 ¼ 9b i 2 b þ 3b ¼ þ 2b j 2 þ 2b ¼ 2( þ b) 4 Write simplified lgebri expression for the perimeter of eh shpe. 2m 3p 2m b 4d 4d 4d 2x 2x 3p 4d 5y d p + 2 e f 9 p p 2p 1 40h 41h h 4 5 Drw tringle nd write lgebri expressions for its side lengths so tht it hs perimeter of 7d þ 6. 6 Drw retngle nd find lgebri expressions for its length nd width so tht it hs perimeter of Worksheet Perimeter nd re 3-02 Multiplying nd dividing terms MAT09NAWK10028 Homework sheet Algebr 1 MAT09NAHS10009 Summry When multiplying nd dividing terms, multiply or divide the numbers first, then the vribles. Exmple 3 Simplify eh expression n 3 4m b 3d 3 4d 2g 3 3h 3 ( 5f) 2 3 5n 3 4m ¼ n 3 m ¼ 40mn Multiply the numbers nd multiply the vribles. Remember to write the vribles in lphbetil order. 70

6 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 b 3d 3 4d ¼ d 3 d ¼ 12d 2 d 3 d ¼ d 2 2g 3 3h 3 ð 5f Þ ¼ ð 5Þ3 g 3 h 3 f ¼ 30fgh Remember to write the vribles in lphbetil order. Exmple 4 Simplify eh expression. 15b b 3b 15b 3b ¼ b b ¼ 5 b 2 b 4b ¼ b 4b ¼ b b 2 b 4b We n ross out the b in the numertor nd denomintor beuse ¼ 1 nd b b ¼ 1 27f 4 ( 3f) d 5p p Divide the numbers nd divide the vribles. 2 ¼ 3 ¼ Divide the numbers nd divide the vribles. d ¼ 4 27f 4 ð 3fÞ ¼ 27f 3f ¼ f f ¼ 9 5p p ¼ 5pp 15p ¼ 5 p6 p 15 6 p ¼ 1 3 p ¼ p 3 The vribles divide by themselves to give 1. p 2 ¼ pp Exerise 3-02 Multiplying nd dividing terms 1 Simplify eh expression x b 5n 3 3 b 3 5 d t 3 6t e 4 3 3w f 2 3 ( 3p) g 4d 3 3e h 6v 3 4w i 8q 3 8q j ( 5) 3 ( 4) k 2y b l 6w 3 3w 3 2 m (5n) 2 n ( 3r) 2 o (t) 2 See Exmple 3 71

7 Chpter Produts nd ftors See Exmple 4 Worked solutions Multiplying nd dividing terms MAT09NAWS Simplify eh expression. 12x 3 b 15y y d 18mn 4 6m e 8vw 4 48vw f g j 18g 3g 75xyz 25x h 25yz 4 5y i b 16b 9p 45pq k 10r r l 12hk 2 4 2hk 3 Simplify 9yx2 24xy2. Selet the orret nswer A, B, C or D. A 3x2 8y 2 B 3x3 8y 3 C 3x 8y 4 Simplify eh expression b 3 5 b 4p q 40n 2 4 5n D 3y 8x d 8b b 2 e 6h 3 2k 3 ( 3) f 2 3 5k 3 ( 4k) g 6d h 7r i ( 33m) r j 9w k 8b 4 2b l 2ef 3 ( 3fe) 45w 5 Use the substitutions p ¼ 5, q ¼ 4 nd r ¼ 3 to test whether eh eqution is orret or inorret p ¼ 12p b q þ q ¼ q 2 2r 3 3r ¼ 6r 2 d 3q 2 3 5q 2 ¼ 15q 2 e 12p 3p ¼ 4 f 7qr qr ¼ 7 g 3r 3 5q ¼ 15qr h p 3 q 3 q ¼ 2pq i 14r 2 7r ¼ 2r j 5pq 10p ¼ q 2 6 Write simplified lgebri expression for the re of eh shpe. b 3 2d 5m 5h 5m k d e 3 f 2k 4p 2 4k 3 6y 72

8 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 7 Find n lgebri expression for the volume of eh retngulr prism. b 2m 5d 6k 2w w 2m 2m 3t 4y 3-03 Adding nd subtrting lgebri frtions Puzzle sheet Algebri frtions MAT09NAPS00010 Animted exmple Summry To dd or subtrt frtions, onvert them (if needed) so tht they hve the sme denomintor, then simply dd or subtrt the numertors. Adding nd subtrting frtions MAT09NAAE00004 Exmple 5 Simplify eh expression. r 3 þ r 3 d 3h 4 þ 2h 5 b 7m 3m e 5m 6 2m 3 k 3 þ k 2 f 3 4 þ 7 10 Video tutoril Adding nd subtrting lgebri frtions MAT09NAVT10004 b r 3 þ r 3 ¼ 2r 3 7m 3m ¼ 4m 10 ¼ 2m 5 k 3 þ k 2 ¼ 2 3 k þ 3 3 k ¼ 2k 6 þ 3k 6 Common denomintor ¼ ¼ 6 ¼ 5k 6 73

9 Chpter Produts nd ftors d e 3h 4 þ 2h 5 ¼ 5 3 3h þ 4 3 2h ¼ 15h 20 þ 8h 20 ¼ 23h 20 5m 6 2m 3 ¼ 15m 18 ¼ 3m 18 ¼ m 6 12m 18 or 5m 6 2m 3 ¼ 5m 6 4m 6 The lowest ommon denomintor is 6. ¼ m 6 f 3 4 þ 7 10 ¼ þ ¼ 15 þ Exerise 3-03 Adding nd subtrting lgebri frtions See Exmple 5 1 Simplify eh expression. w 4 þ 2w b 4k 4 8 þ 7k 8 e 4 q þ 5 q f 5 3d 2 3d i 4t 3 þ s 11y 7y j 3 2h 2h m p z þ 3 z n 5u 3u 8g 8g q 5 6 þ 5 r 5 6 2d 1 2d g 7m 2m r w þ w r k 6 5 o 4 9f 1 9f s 7 5k þ 13 5k d x 3 þ 2x 3 h 5 2 l p t 8 5e þ 7 5e 7e 3e Simplify eh expression. x 3 þ x 4 b s 3 s 7 h 5 þ h 3 d m 7 m 2 e w 4 þ w 5 i m 3 4 þ 2 7 f 5t 2t 4 5 j 2d 11 r 3 n 7e 2e 8 3 g 2p 5 þ p 3 k 3h 5 þ 2 3 o m 2 n 7 h 5r 5r 2 3 l 5 6 þ 4w 5 p 2k 5 þ m 6 74

10 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum Multiplying nd dividing lgebri frtions Puzzle sheet Algebri frtions puzzle MAT09NAPS10029 Summry Puzzle sheet Upside-down frtions MAT09NAPS00018 To multiply frtions, nel ny ommon ftors, then multiply the numertors nd denomintors seprtely To divide by frtion b, multiply by its reiprol b Exmple 6 Simplify eh produt. 3 d 3 4 b 4 k 3 3k r 3 8r 11 3 d 3 4 ¼ d 3 ¼ 12 d b 4 k 3 3k 16 ¼ k k 16 4 ¼ r 3 8r 11 ¼ 5 12r 3 8r 4 11 ¼ ¼ Exmple 7 Simplify eh quotient. 3 h 4 4 k b xy 5 4 3x 25 3 h 4 4 k ¼ 3 h 3 k 4 ¼ 3k 4h b xy 5 4 3x 25 ¼ 1 6 xy x 1 ¼ 5y 3 75

11 Chpter Produts nd ftors Exerise 3-04 Multiplying nd dividing lgebri frtions See Exmple 6 1 Simplify eh produt. w b s 5 3 t 4 3 h 3 5 6k d 4 m 3 3 n e l f f 1 v 3 2 3v See Exmple 7 Worked solutions Multiplying nd dividing lgebri frtions MAT09NAWS g 2 x 3 3 x j 4d 9 3 d 16 m u 10 2 Simplify eh quotient. r 2 4 r 5 d q g 3 e 4 5 e j h k 4 k h m t 3 4 3t 5u 3 Simplify eh expression. 3p 2 4 p2 4 d 5 2g 4 g 2 g b j 5p pt m 5ty k 3 5ky t h b 3 d k 5p p n u u b m 6 4 n 3 e 3y 5 4 2y d h b 6 k 8w 3x 4 2w 9x n 3e 7g 4 e 14g i d e 3 g e l 4 k 3 3 5k o 3z r 3 2r 9dz h 2 4 h 8 4t f 9 4 3t 5 i 5m n 4 2m 3n l 3s 4 4 6s 11 o xh 5 4 3h 15 b 2w 7 4 5w 8y y 2 e xy z 3 5 y f b 3 b 3 b 3 h 4t i 5mn 2d 3 4d n mn k s 3 4 5s 2 3 3s l 7 7 h h n 6 r 3 5r o 6f yh q qf Mentl skills 3A Mths without lultors Multiplying nd dividing by 5, 15, 25 nd 50 It is esier to multiply or divide number by 10 thn by 5. So whenever we multiply or divide number by 5, we n double the 5 (to mke 10) nd then djust the first number. 1 Study eh exmple. To multiply by 5, hlve the number, then multiply by ¼ ðor Þ 2 ¼ ¼ 90 76

12 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 b To multiply by 50, hlve the number, then multiply by ¼ ðor Þ 2 ¼ ¼ 1300 To multiply by 25, qurter the number, then multiply by ¼ ðor Þ 4 ¼ ¼ 1100 d To multiply by 15, hlve the number, then multiply by ¼ ðor Þ 2 ¼ ¼ 120 e To divide by 5, divide by 10 nd double the nswer. We do this beuse there re two 5s in every ¼ ¼ ¼ 28 f To divide by 50, divide by 100 nd double the nswer. This is beuse there re two 50s in every ¼ ¼ ¼ 8 g To divide by 25, divide by 100 nd multiply the nswer by 4. This is beuse there re four 25s in every ¼ ¼ ¼ 24 h To divide by 15, divide by 30 nd double the nswer. This is beuse there re two 15s in every ¼ ¼ ¼ 16 2 Now evlute eh expression b d e f g h i j k l m n o p q r s t

13 Chpter Produts nd ftors Skillsheet Algebr using digrms MAT09NASS10013 Worksheet Algebr review MAT09NAWK Expnding expressions Summry The distributive lw for expnding n expression Multiply eh term inside the brkets by the term outside (b þ ) ¼ b þ (b ) ¼ b Exmple 8 Expnd eh expression. 4(5 þ d) b 2w(3w þ 10) 4(5 + d) = d = d multiply eh term inside the brkets by the term outside (b þ ) ¼ b þ b 2w(3w + 10) = 2w 3w + 2w 10 = 6w w Exmple 9 Expnd eh expression. 3g(h 2) b r(10 4r) 3g(h 2) = 3g h 3g 2 = 3gh 6g multiply eh term inside the brkets by the term outside (b ) ¼ b b r(10 4r) = r 10 r 4r = 10r 4r 2 78

14 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Exmple 10 Expnd eh expression. (y þ 4) b 3(2m 7) (y + 4) = 1(y + 4) = 1 y + ( 1) 4 = y + ( 4) = y 4 ( ) is the sme s 1 3 () b 3(2m 7) = 3 2m ( 3) 7 = 6m ( 21) = 6m + 21 Exmple 11 Expnd nd simplify by olleting like terms. 4(3p þ 2) 5p b 7(n 3) þ n(n 1) 4(3p + 2) 5p = 4 3p p = 12p + 8 5p = 7p + 8 Expnding Colleting like terms to simplify. b 7(n 3) + n(n 1) = 7 n n n n 1 = 7n 21 + n 2 n = n 2 + 7n n 21 = n 2 + 6n 21 It s neter to ple n 2 t the front. Colleting like terms to simplify. 79

15 Chpter Produts nd ftors Exerise 3-05 Expnding expressions See Exmple 8 See Exmple 9 Worked solutions Expnding expressions MAT09NAWS10011 See Exmple 10 See Exmple 11 1 Expnd eh expression. 4(h þ 6) b 5(3 þ t) 3(2b þ 7) d 5(8 þ 3x) e x(x þ 9) f p(5 þ p) g 3r(2r þ 1) h 4y(1 þ 3y) i 10e(2e þ 4f) 2 Expnd eh expression. 3(t 2) b 7(5 d) 8(8g 3) d w(w 7) e ( 1) f 6h(3h 1) g 4x(3x 1) h 5x(2x 4y) i 3(4b 7) 3 Use the substitution x ¼ 5 to test whether eh eqution is orret or inorret. 3(x þ 3) ¼ 3x þ 6 b 7(x 2) ¼ 7x 14 x(4 2x) ¼ 4x 2x 2 4 Expnd eh expression. ( 5) b ( þ 5) 2(x þ 6) d 2(x 6) e (11 þ w) f (11 w) g y(y þ 9) h x(8 z) i 4t(t 8) 5 Expnd 5u(8 þ 2u). Selet the orret nswer A, B, C or D. A 3u 3u 2 B 40u þ 10u 2 C 40u 10u 2 D 40u 3u 2 6 Expnd nd simplify by olleting like terms. 5(3m þ 2) þ 4m b 3(1 5e) þ 6e 4w 2(5 þ 2w) d 8 5(2x 7) e t(t þ 4) þ 3(t þ 4) f 4(3 þ h) þ h(7 2h) g 3x(2x þ 5) þ 4(2x þ 5) h v(2v þ 3) 6(v þ 1) i 3(1 2w) w(2 w) j 2y(3y 7) 5(3y 7) Skillsheet Ftorising using digrms MAT09NASS10015 Homework sheet Algebr 2 MAT09NAHS10010 Puzzle sheet Ftorising puzzle MAT09NAPS Ftorising expressions Summry The highest ommon ftor (HCF) of two or more terms is the lrgest term tht is ftor of ll of the terms. To find the HCF of lgebri terms: find the HCF of the numbers find the HCF of the vribles multiply them together 80

16 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Exmple 12 Find the highest ommon ftor (HCF) of eh pir of terms. 20x 2 nd 15xy b 16 nd 12b Find the HCF of the numbers nd the HCF of the vribles. Their produt is the HCF of the expression. The HCF of 20 nd 15 is 5. The HCF of x 2 nd xy is x, sine it is the lrgest ommon prt of x 2 nd xy. [ The HCF of 20x 2 nd 15xy ¼ 5 3 x ¼ 5x b The HCF of 16 nd 12 is 4. There is no HCF of the vribles beuse b is in 12b but not in 16. [ The HCF of 16 nd 12b is 4. Skillsheet HCF by ftor trees MAT09NASS10014 Skillsheet Ftorising using digrms MAT09NASS10015 Ftorising lgebri expressions When 4(2y þ 5) is expnded, the nswer is 8y þ 20. Ftorising is the reverse of expnding. Ftoristion breks n expression into ftors. To ftorise 8y þ 20, we tke out the gretest ommon divisor nd insert brkets. The result is 4(2y þ 5). The ftors re 4 nd (2y þ 5). Summry Ftorising n expression Find the HCF of the terms nd write it outside the brkets Divide eh term by the HCF nd write the nswers inside the brkets b þ ¼ (b þ ) b ¼ (b ) To hek tht the ftorised nswer is orret, expnd it Expnd (remove brkets) 4(2y + 5) 8y + 20 Ftorise (insert brkets) 81

17 Chpter Produts nd ftors Video tutoril Ftorising expressions MAT09NAVT10005 Exmple 13 Ftorise eh expression. 8y þ 16 b 25b 2 20b v(4 þ w) þ 2(4 þ w) The HCF of 8y nd 16 is 8. ) 8y þ 16 ¼ 8 3 y þ ¼ 8ðy þ 2Þ b The HCF of 25b 2 nd 20b is 5b. ) 25b 2 20b ¼ 5b 3 5b 5b 3 4 ¼ 5bð5b 4Þ The HCF of v(4 þ w) þ 2(4 þ w) is (4 þ w). ) vð4 þ wþþ24þ ð wþ ¼ ð4 þ wþ3 v þ ð4 þ wþ3 2 ¼ ð4 þ wþðv þ 2Þ Rewrite the expression using the HCF 8. Write the HCF t the front of the brkets. Rewrite the expression using the HCF 5b. Write the HCF t the front of the brkets. Ftorising with negtive terms Exmple 14 Ftorise eh expression. x 2 þ 3x b b When ftorising expressions tht begin with negtive term, we use the negtive HCF. The gretest negtive ommon divisor of x 2 nd 3x is x. ) x 2 þ 3x ¼ ð xþ3 x þð xþ3 ð 3Þ ¼ ð xþðx þð 3ÞÞ ¼ xx ð 3Þ b The gretest negtive ommon divisor of nd b is. ) b ¼ 31þð Þ3b ¼ ð1þbþ ( x) 3 ( 3) ¼þ3x ( ) 3 b ¼ b 82

18 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Summry Ftorising with negtive terms Find the negtive HCF of the terms nd write it outside the brkets Divide eh term by the HCF nd write the nswers inside the brkets b ¼ (b þ ) b þ ¼ (b ) To hek tht the ftorised nswer is orret, expnd it Exerise 3-06 Ftorising expressions 1 Find the HCF of the following terms. 12, 6y b 8 2,24 20b, 15 d 6, 9p 2 e 24mn, m f, 2 g 18pq, 12p 2 h 24w 2,16w i 3(x 5), x(x 5) 2 Copy nd omplete eh ftoristion. 18p 2 þ 24 ¼ 6( þ ) b 9m 2 þ 3m ¼ 3m( þ ) 20d 30d 2 ¼ 10d( ) d 6x 2 y 8xy ¼ 2xy( ) e x(x 2) þ 5(x 2) ¼ (x 2)( þ ) f n(3 þ n) 3(3 þ n) ¼ (3 þ n)( ) g h(h þ 4) 2(h þ 4) ¼ ( þ )(h 2) h v(1 w) þ (1 w) ¼ ( þ )(v þ 1) 3 Ftorise eh expression. 24x þ 30 b 36 þ 27 16g 2 64 d xy þ y e mn 3n f 6p þ pq g x 2 þ x h 2y y 2 i 3d 2 þ 6d j 16r 2 12r k 6t 2 þ 27t l 36p 2 108p m 12x 2 y 16x n 18p 2 þ 16pr o 4m 2 n 4mn 2 p 14b þ 21b q 28vw 21v 2 w 2 r 45rt þ 54r 2 t s 36pq 2 r 144pr t 48x 2 y þ 64xy 2 u 75g 3 h 2 125gh 4 Ftorise eh expression. ( 3) þ 6( 3) b t(8 þ t) 3(8 þ t) b(b þ 5) 2(b þ 5) d x(2 y) 6(2 y) e 5( 7) þ b( 7) f s( þ 3) þ 4( þ 3) g 3p(g þ 5) 2(g þ 5) h 5(2 3m) þ 2n(2 3m) i r(8 þ r) þ (8 þ r) j (y 6) y(y 6) 5 Ftorise eh expression using the negtive HCF. 4q 8 b 9u þ g d 18 þ 12 e n 1 f n þ 1 g y 2 9y h 6t þ 10t 2 i 3 2 6b j p þ q k 20e 2 22e l 9m þ 3m 2 6 Ftorise 10kr þ 4rn. Selet the orret nswer A, B, C or D. A 2r(5k 4n) B 2r(5k 2n) C 5r(2k 2n) D 2r(5k þ 2n) 7 Ftorise eh expression. 1 2 p þ 1 2 g b 7g þ 9g x2 1 4 xy d 8 þ 12y þ 4 e 20xy 10x 2 þ 5y f t 2 m þ tm 2 þ tm g 7y(m þ 2) p(m þ 2) h 10p 2 10pq i 5(x y) þ 10 2 xy 10 2 y 2 See Exmple 12 See Exmple 13 See Exmple 14 83

19 Chpter Produts nd ftors Worksheet Are digrms MAT09NAWK10031 Worksheet Binomil produts MAT09NAWK10032 Homework sheet Algebr 3 MAT09NAHS Expnding binomil produts (x þ 5) nd (x 1) re lled binomil expressions beuse eh expression hs extly two terms (binomil ¼ 2 terms ). (x þ 5)(x 1) is lled binomil produt beuse it is produt (multiplition nswer) of two binomil expressions. Expnding binomil produts using n re digrm Exmple 15 Puzzle sheet Expnding brkets MAT09NAPS00009 Expnd eh binomil produt using n re digrm. ( þ 2)( þ 5) b (n þ 4)(n 3) Tehnology GeoGebr: Binomil expnsion MAT09NATC00003 Tehnology worksheet Exel worksheet: Expnding binomils MAT09NACT00021 Drw n re digrm (retngle) with length ( þ 2) nd width ( þ 5) nd divide the digrm into 4 smller retngles Tehnology worksheet Exel spredsheet: Expnding binomils Find the re of eh smller retngle MAT09NACT Expnd ( þ 2)( þ 5) by dding the res of the 4 retngles. ð þ 2Þð þ 5Þ ¼ 2 þ 2 þ 5 þ 10 ¼ 2 þ 7 þ 10 b Drw n re digrm with length (n þ 4) nd width (n 3), then inrese the width to n by dding 3. This retes 4 retngles. Expnding Simplifying by olleting like terms. n + 4 n 4 n 3 n 3 84

20 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 The 2 shded upper retngles together give the produt (n þ 4)(n 3). To find its re, we subtrt the res of the 2 lower retngles from the re of the whole retngle. n 3 n n + 4 n 4 (n + 4)(n 3) 3 3n 12 ðn þ 4Þðn 3Þ ¼ nnþ ð 4Þ 3n 12 ¼ n 2 þ 4n 3n 12 ¼ n 2 þ n 12 Whole retngle 2 lower retngles Expnding Simplifying by olleting like terms. Expnding binomil produts lgebrilly When expnding binomil produt lgebrilly, eh term in the first binomil is multiplied by eh term in the seond binomil to give four terms, whih re olleted nd simplified. Exmple 16 Expnd eh binomil produt. (x þ 5)(x þ 9) b (k þ 3)(k 7) Video tutoril Expnding binomil produts MAT09NAVT10006 ðx þ 5Þðx þ 9Þ ¼ xxþ ð 9Þþ5ðx þ 9Þ ¼ x 2 þ 9x þ 5x þ 45 ¼ x 2 þ 14x þ 45 b ðk þ 3Þðk 7Þ ¼ kk ð 7Þþ3ðk 7Þ ¼ k 2 7k þ 3k 21 ¼ k 2 4k 21 Eh term in (x þ 5) is multiplied by (x þ 9). Expnding to mke 4 terms. Adding 9x nd 5x. Eh term in (k þ 3) is multiplied by (k 7). Expnding to mke 4 terms. Adding 7k nd 3k. Summry The distributive lw for expnding binomil produt Multiply eh term in the first binomil by eh term in the seond binomil. ( þ b)( þ d) ¼ þ d þ b þ bd + d + b b b d d bd 85

21 Chpter Produts nd ftors One wy of remembering whih pirs of terms to multiply together in binomil produt is lled the FOIL method, s shown below. F mens multiply the first terms: k 3 k ¼ k 2 O mens multiply the outside terms: k 3 ( 7) ¼ 7k I mens multiply the inside terms: 3 3 k ¼ 3k L mens multiply the lst terms: 3 3 ( 7) ¼ 21 F O ( k + 3)( k 7) = k 2 7 k + 3 k 21 I = k 2 4 k 21 L Video tutoril Expnding binomil produts MAT09NAVT10006 Video tutoril Expnding binomil produts MAT09NAVT00004 Exmple 17 Expnd eh binomil produt. (x 6)(4x þ 2) b (3t 1)(2t 5) ðx 6Þð4x þ 2Þ ¼ xð4x þ 2Þ 64x ð þ 2Þ ¼ 4x 2 þ 2x 24x 12 ¼ 4x 2 22x 12 b ð3t 1Þð2t 5Þ ¼ 3tð2t 5Þ 12t ð 5Þ ¼ 6t 2 15t 2t þ 5 ¼ 6t 2 17t þ 5 Expnding Simplifying Expnding Simplifying Exerise 3-07 Expnding binomil produts See Exmple 15 1 Expnd eh binomil produt using the given re digrm. (x þ 10)(x þ 8) b ( þ 3)( þ 6) x 10 3 x 8 6 (d þ 2)(2d þ 5) d (4h þ 1)(3h þ 7) d 2 4h 1 2d 3h

22 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 2 Expnd eh binomil produt using the given re digrm. (Find the shded re.) (w þ 5)(w 1) b (x þ 9)(x 6) w 5 x 9 w 1 w 1 x 6 x 6 (t 8)(t þ 4) d (n 5)(n 7) t 8 t 8 n 5 n 5 t 4 n 7 n 7 3 Drw n re digrm for eh binomil produt nd use it to expnd the produt. (2y þ 5)(y þ 3) b (r þ 5)(3r þ 2) (5k þ 3)(4k þ 1) d (w 5)(w þ 3) e (v þ 4)(3v 1) f (p 3)(p 5) 4 Expnd eh binomil produt lgebrilly. (n þ 6)(n þ 3) b (t þ 6)(t þ 5) (p þ 10)(p 10) d (x 8)(x þ 3) e (b 2)(9 þ b) f (u 8)(u 7) g (15 r)(r þ 1) h ( 10)( 9) i (5 )(3 ) j (t 1)(t þ 2) k (y 4)(y þ 10) l (n 9)(11 þ n) 5 Expnd (b þ 7) 2. Selet the orret nswer A, B, C or D. A b 2 þ 49 B b 2 þ 49b C b 2 þ 7b þ 49 D b 2 þ 14b þ 49 6 Expnd eh binomil produt. (2x þ 5)(x þ 3) b (3e þ 2)(4e þ 5) (10 þ 3p)(p 1) d (7d 2) (7d 2) e (2f 2)(3f þ 5) f (4m 5)(5 þ 3m) g (3 4h)(2 þ 5h) h (4p 5)(2p 7) i (2m 3)(4 5m) j (3t þ 5)(2t 1) k (5y 5)(5y þ 5) l (6 2)(2 6) 7 A retngulr brbeue plte hs length of 100 m nd width of 75 m. The length nd width re both inresed by x m. Write n expression for the new length of the plte. b Write n expression for the new width of the plte. Hene find simplified expression for the new re of the plte. d By how muh hs the re of the plte inresed? e If x ¼ 0.1, find the inrese in the re of the plte. See Exmple 16 See Exmple 17 Worked solutions Expnding binomil produts MAT09NAWS

23 Chpter Produts nd ftors 8 A fmily room in house is to be extended. The room is 4 metres long nd 3 metres wide. The length is to be inresed by x metres nd the width by y metres. Write down expressions for the new length nd width. b Write down binomil expression for the new re of the room. Expnd nd simplify your expression for the re. d By how muh hs the re of the room inresed? Investigtion: Expnding perfet squres There is speil pttern when you expnd binomil by itself, for exmple, (y þ 5)(y þ 5). This is lled the squre of binomil or perfet squre. 1 Expnd nd simplify the perfet squre (y þ 5) 2 if (y þ 5) 2 ¼ (y þ 5)(y þ 5). b How mny terms re there in your nswer? The terms in the binomil (y þ 5) re y nd 5. The first term of your nswer in prt is y 2, whih is the squre of y. How re the other terms in the nswer relted to y nd 5? 2 Expnd nd simplify eh perfet squre. (k þ 3) 2 b (m þ 7) 2 (p þ 2) 2 3 For eh of the expnsions in question 2: How mny terms re there? b Desribe how eh is relted to the two terms of the perfet squre. Compre nd disuss your results with other students. 4 Expnd nd simplify eh perfet squre. (t 1) 2 b (g 6) 2 (d 5) 2 5 For eh of the expnsions in question 4: How mny terms re there? b Desribe how eh is relted to the two terms of the perfet squre. Compre nd disuss your results with other students. 6 Desribe the expnsion of the perfet squre (first term þ seond term) 2 by opying nd ompleting this sttement: The squre of binomil is equl to the squre of the first term plus double the produt of the two terms plus 7 Find formul for ( þ b) 2. b Chek this formul by expnding ( þ b) 2 using n re digrm. Find formul for ( b) 2. d Chek this formul by expnding ( b) 2 using n re digrm. 88

24 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum Perfet squres 16, 49, v 2 nd (y þ 5) 2 re lled perfet squres beuse they re squre numbers. The rule for expnding the perfet squre of binomil is: Stge 5.3 Puzzle sheet Ftorominoes MAT09NAPS10033 Summry ( þ b) 2 ¼ 2 þ 2b þ b 2 ( b) 2 ¼ 2 2b þ b 2 Proof: ð þ bþ 2 ¼ ð þ bþð þ bþ ¼ þ ð bþþbþ ð bþ ¼ 2 þ b þ b þ b 2 ¼ 2 þ 2b þ b 2 ð bþ 2 ¼ ð bþð bþ ¼ ð bþ b ð bþ ¼ 2 b b þ b 2 ¼ 2 2b þ b 2 Exmple 18 Copy nd omplete the expnsion of eh perfet squre. (x þ 4) 2 ¼ x 2 þ þ 16 b (y 6) 2 ¼ y 2 12y þ (5g þ 9) 2 ¼ þ 45g þ 81 d (3d 5) 2 ¼ 9d 2 þ 25 In the expnsion, ¼ 2 3 x 3 4 ¼ 8x ) ðx þ 4Þ 2 ¼ x 2 þ 8x þ 16 Doubling the produt of the two terms. b In the expnsion, ¼ 6 2 ¼ 36 ) ðy 6Þ 2 ¼ y 2 12y þ 36 The seond term squred. ¼ ð5gþ 2 ¼ 25g 2 ) ð5g þ 9Þ 2 ¼ 25g 2 þ 45g þ 81 The first term squred. d ¼ 2 3 3d 3 5 ¼ 30d ) ð3d 5Þ 2 ¼ 9d 2 þ 30d þ 25 Doubling the produt of the two terms. 89

25 Chpter Produts nd ftors Stge 5.3 Exmple 19 Expnd eh perfet squre. (n 5) 2 b (k þ 7) 2 (3y 8) 2 ðn 5Þ 2 ¼ n n 3 5 þ 5 2 ¼ n 2 10n þ 25 b ðk þ 7Þ 2 ¼ k 2 þ 2 3 k 3 7 þ 7 2 ¼ k 2 þ 14k þ 49 ð3y 8Þ 2 ¼ ð3yþ y 3 8 þ 8 2 ¼ 9y 2 48y þ 64 1st term squred double produt þ 2nd term squred Exerise 3-08 Perfet squres See Exmple 18 See Exmple 19 1 Copy nd omplete the expnsion of eh perfet squre. (x þ 10) 2 ¼ x 2 þ þ 100 b (m 8) 2 ¼ 16m þ 64 (p t) 2 ¼ p 2 2pt þ d (h þ 4) 2 ¼ h 2 þ 16 e (k 9) 2 ¼ k 2 þ 81 f (8 þ 5f ) 2 ¼ 64 þ 25f 2 g (2d þ 3) 2 ¼ þ þ 9 h (6 þ 1) 2 ¼ þ 12 þ 2 Expnd eh perfet squre. (m þ 9) 2 b (u þ 3) 2 (y 6) 2 d (8 þ k) 2 e (5 h) 2 f (7 þ k) 2 g (f þ 20) 2 h (q 11) 2 i (10 þ t) 2 j (x w) 2 k ( þ g) 2 l (2m 3) 2 m (5x 6) 2 n (9 þ 2) 2 o (3e 4) 2 p (5 þ 7b) 2 q (4 5p) 2 r (11 2) 2 s (10g þ 3) 2 t (3k þ 11) 2 u (5 þ 2v) 2 3 Expnd eh perfet squre. (7h þ 2k) 2 b (8 3y) 2 (xy þ z) 2 d 1 þ 1 2 y e t t f w þ w 4 Use expnsion to evlute eh squre number without using lultor ¼ (20 þ 1) 2 b 45 2 ¼ (40 þ 5) ¼ (30 1) 2 d 59 2 ¼ (60 1) 2 e ¼ (100 þ 2) 2 f 98 2 ¼ (100 2) 2 90

26 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 5 A 10 m squre sheet of rdbord hs squre of length x m ut from eh orner. It is then folded to form squre-bsed prism. x 10 x Stge 5.3 Worked solutions Perfet squres x x MAT09NAWS x b b x x x x Why is the length of the squre bse (b) equl to (10 2x) m? b Find the re of the squre bse. Find the re of one side fe of the prism. d Hene show tht the surfe re of the prism is (100 4x 2 )m 2. Investigtion: Squring number ending in 5 Study this mentl short ut for squring number ending in 5. To evlute 35 2, lulte ¼ 12, dd 25 to the end: 35 2 ¼ To evlute 75 2, lulte ¼ 56, dd 25 to the end: 75 2 ¼ To evlute 105 2, lulte ¼ 110, dd 25 to the end: ¼ Let n stnd for the tens digit of the number ending in 5 being squred. Expnd (10n þ 5) 2 nd investigte why the bove method works. Mentl skills 3B Mths without lultors Multiplying by 9, 11, 99 nd 101 We n use expnsion when multiplying by number ner 10 or ner Study eh exmple ¼ 25 3 ð10 þ 1Þ ¼ þ ¼ 250 þ 25 ¼ 275 b ¼ 14 3 ð10 1Þ ¼ ¼ ¼

27 Chpter Produts nd ftors ¼ 32 3 ð10 þ 2Þ ¼ þ ¼ 320 þ 64 ¼ 384 e ¼ 27 3 ð100 þ 1Þ ¼ þ ¼ 2700 þ 27 ¼ 2727 d ¼ 7 3 ð100 1Þ ¼ ¼ ¼ 693 f ¼ 18 3 ð10 2Þ ¼ ¼ ¼ Now evlute eh produt b d e f g h i j k l m m o p Stge 5.3 Investigtion: Expnding sums by differenes There is lso speil pttern when you multiply the sum of two terms by the differene of those two terms, for exmple, (x þ 9)(x 9). 1 Expnd nd simplify (x þ 9)(x 9). b How mny terms re there in your nswer? How re the terms in the nswer relted to x nd 9? 2 Expnd nd simplify eh sum by differene. (k þ 2)(k 2) b (m þ 5)(m 5) (p 6)(p þ 6) 3 Desribe the expnsion of (first term þ seond term)(first term seond term) by opying nd ompleting this sttement: The produt of sum by differene is equl to the squre of the first term 4 Find formul for ( þ b)( b). b Chek this formul by expnding ( þ b)( b) using n re digrm. Worksheet Speil produts MAT09NAWK00034 Puzzle sheet Enough time MAT09NAPS Differene of two squres The rule for expnding ( þ b)( b) is: Summry ( þ b)( b) ¼ 2 b 2 The nswer is lled the differene of two squres. 92

28 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Proof: ð bþð þ bþ ¼ þ ð bþ bþ ð bþ ¼ 2 þ b b b 2 ¼ 2 b 2 þ b b ¼ 0 Stge 5.3 When the sum of two terms is multiplied by their differene, the nswer is the squre of the first term minus the squre of the seond term (the differene of two squres). Exmple 20 Expnd eh expression. (d þ 3)(d 3) b (2 þ r)(2 r) (7x þ 2)(7x 2) d (4k 5p)(4k þ 5p) ðd þ 3Þðd 3Þ ¼ d ¼ d 2 9 ð7x þ 2Þð7x 2Þ ¼ ð7xþ ¼ 49x 2 4 b d ð2 þ rþð2 rþ ¼ 2 2 r 2 ð4k 5pÞð4k þ 5p ¼ 4 r 2 Þ ¼ ð4k Þ 2 ð5pþ 2 ¼ 16k 2 25p 2 Exerise 3-09 Differene of two squres 1 Expnd eh expression. (m þ 5)(m 5) b ( 10)( þ 10) ( þ 12)( 12) d (6 y)(6 þ y) e (8 m)(8 þ m) f (p þ 1)(p 1) g (5 þ e)(5 e) h (v þ 11)(v 11) i (w 3)(w þ 3) j (x 10)(x þ 10) k (q þ 7)(q 7) l (9 g)(9 þ g) m (b 2)(b þ 2) n (15 r)(15 þ r) o (d þ 13)(d 13) 2 Expnd eh expression. (2h 3)(2h þ 3) b (5r þ 4)(5r 4) (5b þ 8)(5b 8) d (4p 7)(4p þ 7) e (3 8k)(3 þ 8k) f (7x 5)(7x þ 5) g (2 þ 9m)(2 9m) h (9k 4l)(9k þ 4l) i (7n þ 8m)(7n 8m) j (4g 5h)(4g þ 5h) k (7u þ 3w)(7u 3w) l (11 þ 3b)(11 3b) m t þ 1 t t 1 t n w 3 2 w 3 þ 2 o 1 1 r 1 þ 1 r 3 Susn s ge is p yers. Wht ws Susn s ge lst yer? b Wht will Susn s ge be next yer? Write n expression for (Susn s ge lst yer) 3 (Susn s ge next yer). d If (Susn s ge lst yer) 3 (Susn s ge next yer) is equl to 48, wht is Susn s ge? 4 By expressing s (30 þ 1)(30 1), use the differene of two squres to find the vlue of See Exmple 20 Worked solutions Differene of two squres MAT09NAWS

29 Chpter Produts nd ftors Stge 5.3 Worked solutions Differene of two squres MAT09NAWS Use the method of question 4 to evlute eh expression b d Use sum by differene to evlute eh expression b d e f A squre sheet of pper of length x m hs smller squre of length y m ut from one orner. It is then ut on the digonl AB nd rerrnged to form retngle s shown. w A x A x B y l y B Find n expression for: i the length (l) of the retngle ii the width (w) of the retngle iii the re of the retngle. b The re of the retngle should be the sme s the re of the squre on the left minus the re of the smller squre. Find n expression for this re. Wht does this prove? Worksheet Mixed expnsions 3-10 Mixed expnsions MAT09NAWK00035 Exmple 21 Expnd nd simplify eh expression. (4r þ 5)(1 2r) b (7 þ 9x) 2 (2d 10)(2d þ 10) d ( þ 6)( 6) þ ( þ 12)( þ 3) e (m 2) 2 (m 2)(m þ 2) ð4r þ 5Þð1 2rÞ ¼ 4rð1 2rÞþ51 ð 2rÞ ¼ 4r 8r 2 þ 5 10r ¼ 8r 2 6r þ 5 94

30 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 b ð7 þ 9xÞ 2 ¼ 7 2 þ x þ ð9xþ 2 ¼ 49 þ 126x þ 81x 2 ¼ 81x 2 þ 126x þ 49 ð2d 10Þð2d þ 10Þ ¼ ð2dþ ¼ 4d d ð þ 6Þð 6Þþð þ 12Þð þ 3Þ ¼ 2 36 þ 2 þ 3 þ 12 þ 36 ¼ 2 2 þ 15 e ðm 2Þ 2 ðm 2Þðm þ 2Þ ¼ m 2 4m þ 4 m 2 4 ¼ m 2 4m þ 4 m 2 þ 4 ¼ 4mþ8 Stge 5.3 Exerise 3-10 Mixed expnsions 1 Expnd nd simplify eh expression. (2m 1)(2m þ 1) b (y þ 4) (5y 3) (2k 7) 2 d (d þ 9) 2 e (2e 1)(e þ 1) f (5 þ 4)(5 4) g (2 p)(p 2) h (10 6y)(10 þ 6y) i (h 3z) 2 j (2x 3)(y þ 3) k (11 4b)(11 þ 4b) l u 1 2 u 2 Expnd nd simplify eh expression. (m 5)(m þ 5) þ 25 b 6y þ (y 3) 2 þ 9 (3x þ 1)(2 x) þ 2x þ 4 d (d þ 4) 2 8d þ 5 e 16 þ (4k 8)(4k þ 8) f (x y) 2 (x þ y) 2 g 20t (t 2)(t 5) þ t 2 h 2(f 2)(f þ 2) i (2h þ 3) 2 (2h 3)(2h þ 3) j 7xy (2x 3)(y þ 3) 3 Expnd nd simplify eh expression. (8 1)(8 þ 1) 4 2 þ 1 b (n þ 1) 2 þ 2n þ 3 3(4 t)(4 þ t) þ (t 12)(t þ 4) d (2m n) 2 þ (2m þ n) 2 e (x 2)(x þ 3) (x 2)(x þ 2) f 2(b 1) 2 (2b 1) 2 g (y þ 1) 2 þ (y þ 2) 2 þ (y þ 3) 2 h (x 3)(x þ 3) þ (x þ 3) 2 þ (x 3) 2 i (5n þ 3)(5n 3) þ (3n 5)(3n þ 5) j 2( b)( þ b) ( þ b) 2 ( b) 2 See Exmple 21 Worked solutions Mixed expnsions MAT09NAWS Ftorising speil binomil produts Ftorising by grouping in pirs An lgebri expression with four terms n often be ftorised in pirs, tht is, two terms t time, to mke binomil produt. 95

31 Chpter Produts nd ftors Stge 5.3 Exmple 22 Ftorise eh expression. 3 þ 2bd þ 2b þ 3d b 4km þ 6mn 6kp 9np 10xw 6yw 10xt þ 6yt b 3 þ 2bd þ 2b þ 3d ¼ 3 þ 3d þ 2bd þ 2b ¼ 3ðþ dþþ2bðdþ Þ ¼ ð þ dþð3 þ 2bÞ 4km þ 6mn 6kp 9np ¼ 2mð2k þ 3nÞ 3pð2k þ 3nÞ ¼ ð2k þ 3nÞð2m 3pÞ 10xw 6yw 10xt þ 6yt ¼ 25xw ð 3yw 5xt þ 3ytÞ ¼ 25xw ð 5xt 3yw þ 3ytÞ ¼ 25xw ½ ð tþ 3yðw tþš ¼ 2ðw tþð5x 3yÞ Grouping into pirs for ftorising. Ftorising eh pir. Ftorising gin. Ftorising eh pir. Ftorising gin. Ftorising ll terms first. Grouping into pirs for ftorising. Ftorising eh pir. Ftorising gin. Ftorising the differene of two squres You should rell the produt ( þ b)( b) ¼ 2 b 2. If we use this rule in reverse, then the ftors of 2 b 2 re ( b) nd ( þ b). Summry 2 b 2 ¼ ( þ b)( b) Exmple 23 Ftorise eh expression. x 2 4 b 9 16b 2 20d d y 3 y x 2 4 ¼ x ¼ ðx þ 2Þðx 2Þ 20d ¼ 54d 2 2 h i ¼ 5 ð2dþ 2 2 ¼ 52d ð þ Þð2d Þ b 9 16b 2 ¼ 3 2 ð4bþ 2 ¼ ð3 þ 4bÞð3 4bÞ d y 3 y ¼ yy 2 1 ¼ yyþ ð 1Þðy 1Þ 96

32 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Exerise 3-11 Ftorising speil binomil produts Stge Ftorise eh expression. 4b þ 5b þ 4d þ 5d b 2xy 5wy þ 2xt 5wt 9 þ 6b þ 12d þ 8bd d 10x 2 þ 30 þ x 3 þ 3x e 3 2 þ 3b þ 3 þ 3b f 6rt 18wt þ 6rp 18wp g 14e 21 þ 2de 3d h hk h 2 2k þ 2h i 3mn 6m þ pn 2p j 9p 2 27 þ qp 2 3q k fg fh 10g þ 10h l 9kl 12ml þ 9kn 12mn m 2p 2 p 2 þ p n l 3 þ lm 2 3l 2 3m 2 o (x þ 1) þ y(x þ 1) k ky p p( b) 2q( b) þ 3qp 6q 2 2 Ftorise eh expression. w 2 9 b y 2 36 k 2 1 d m e p 2 64 f t g 4e 2 f 2 h 2 9b 2 i 16y 2 1 j 4 b 2 k 25 e 2 l 1 16x 2 m y 2 z 2 n 49 16m 2 o b 2 121d 2 p k 2 q 16 81h 2 r m 2 s n 2 t 121p 2 144q 2 u v 4t w 25h x 1 m 2 n 2 3 Ftorise eh expression b 2 b 7k u 2 d x 3 49x e k 16k 3 f 50q 2 2 g 3d 2 12v 2 h 5t 5 125t 3 i 2 2 b 2 2 j x 2 y 2 x 2 w 2 k 192f 2 108g 2 l 45d See Exmple 22 See Exmple 23 Worked solutions Ftorising speil binomil produts MAT09NAWS10016 m 2x n w 2 o e2 p Ftorise eh expression. e p2 x b2 4 i (p 2q) 2 (2p þ q) 2 j b x v 2 u d 2y2 9 2m2 121 f t 4 81 g 100 n 4 h (x þ y) 2 x 2 x 2 y k ( þ b) 2 ( b) 2 Investigtion: Ftorising qudrti expressions 1 Show tht (x þ 3)(x þ 5) ¼ x 2 þ 8x þ 15 b The qudrti expression x 2 þ 8x þ 15 hs three terms. The oeffiient of x is 8, the number in front of the x. How re the 3 nd 5 in (x þ 3)(x þ 5) relted to the 8? The onstnt term in x 2 þ 8x þ 15 is 15, the number with no x t the end. How re the 3 nd 5 relted to the 15? 2 Expnd (x þ 9)(x þ 2). b Wht is the oeffiient of x? How re 9 nd 2 relted to it? Wht is the onstnt term? How re 9 nd 2 relted to it? 3 Expnd (x þ 8)(x 3). b Wht is the oeffiient of x? How re 8 nd 3 relted to it? Wht is the negtive onstnt term? How re 8 nd 3 relted to it? 97

33 Chpter Produts nd ftors Stge Expnd (x 4)(x 1). b Wht is the negtive oeffiient of x? How re 4 nd 1 relted to it? Wht is the positive onstnt term? How re 4 nd 1 relted to it? 5 In the expnsion of ny binomil produt, how re the oeffiient of x nd the onstnt term relted to the numbers in the binomils? 6 Copy nd omplete: (x þ )(x þ ) ¼ x 2 þ 5x þ 4 b (x þ )(x þ ) ¼ x 2 þ 8x þ 15 (x þ )(x þ ) ¼ x 2 þ 7x þ 12 d (x þ )(x ) ¼ x 2 4x 32 e (x þ )(x ) ¼ x 2 þ 2x 3 f (x )(x ) ¼ x 2 9x þ Ftorising qudrti expressions A qudrti expression suh s x 2 5x þ 7 is lled trinomil beuse it hs three terms. Other exmples of qudrti trinomils re x 2 þ x 15, 2x 2 3x þ 9 nd 4x 2 þ 9x þ 20. The expnsion of (x þ 2)(x þ 4) is x 2 þ 6x þ 8. [ The ftoristion of x 2 þ 6x þ 8is(x þ 2)(x þ 4). Summry In the ftoristion of qudrti trinomil suh s x 2 þ 6x þ 8: eh ftor must hve n x term to give x 2 x 2 + 6x + 8 = (x + 2)(x + 4) 2 þ 4 ¼ 6, whih is the oeffiient of x, the number in front of the x x 2 + 6x + 8 = (x + 2)(x + 4) ¼ 8, whih is the onstnt term with no x. x 2 + 6x + 8 = (x + 2)(x + 4) Exmple 24 Ftorise x 2 þ 11x þ 24. Find the two numbers tht hve sum of 11 nd produt of 24. It is best to test numbers tht hve produt of 24 nd then hek if their sums equl 11: The orret numbers re 8 nd 3. [ x 2 þ 11x þ 24 ¼ (x þ 8)(x þ 3) Pir of numbers Produt Sum 6, ¼ 24 6 þ 4 ¼ 10 2, ¼ 24 2 þ 12 ¼ 14 8, ¼ 24 8 þ 3 ¼ 11 98

34 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Exmple 25 Stge 5.3 Ftorise x 2 5x 6. Find two numbers with sum of 5 nd produt of 6. Sine their produt is 6, one of the numbers must be negtive. Test numbers tht hve produt of 6 nd hek their sums. The orret numbers re 6 nd 1. [ x 2 5x 6 ¼ (x 6)(x þ 1) Pir of numbers Produt Sum þ3, ( 2) ¼ 6 3 þ ( 2) ¼ 1 3, þ ¼ 6 3 þ 2 ¼ 1 þ6, ( 1) ¼ 6 6 þ ( 1) ¼ 5 6, þ ¼ 6 6 þ 1 ¼ 5 Summry To ftorise qudrti trinomils of the form x 2 þ bx þ : find two numbers tht hve sum of b nd produt of use these two numbers to write binomil produt of the form (x )(x ) Exmple 26 Ftorise eh qudrti expression. x 2 þ 6x 16 b m 2 7m 18 y 2 2y þ 1 x 2 þ 6x 16 Find two numbers tht hve produt of 16 nd sum of 6. Sine the produt is negtive, one of the numbers must be negtive. They re þ8 nd 2. [ x 2 þ 6x 16 ¼ (x þ 8)(x 2) b m 2 7m 18 Produt ¼ 18, sum ¼ 7. Sine the produt is negtive, one of the numbers must be negtive. They re 9 nd þ2. [ m 2 7m 18 ¼ (m 9)(m þ 2) y 2 2y þ 1 Produt ¼ 1, sum ¼ 2. Sine the sum is negtive, one of the numbers must be negtive. Sine the produt is positive, both of the numbers must be negtive. They re 1 nd 1. ) y 2 2y þ 1 ¼ ðy 1Þðy 1Þ ¼ ðy 1Þ 2 Note: (y 1) 2 is perfet squre. 99

35 Chpter Produts nd ftors Stge 5.3 Exmple 27 Ftorise eh qudrti expression. 3g 2 þ 12g 36 b 48 8p p 2 3g 2 þ 12g 36 ¼ 3 g 2 þ 4g 12 ¼ 3ðg 2Þðg þ 6Þ b 48 8p p 2 ¼ p 2 8p þ 48 ¼ 1 p 2 þ 8p 48 ¼ p ð þ 12Þðp 4Þ Tking out the HCF of 3 first. Produt ¼ 12, sum ¼ 4 Rerrnging the terms to mke the p 2 term first. Tking out ommon ftor of 1. Produt ¼ 48, sum ¼ 8 Exerise 3-12 Ftorising qudrti expressions See Exmple 24 See Exmple 25 See Exmple 26 See Exmple 27 1 Find two numbers whose: produt is 6 nd their sum is 7 b produt is 12 nd their sum is 1 produt is 15 nd their sum is 2 d produt is 12 nd their sum is 7 e produt is 20 nd their sum is 9 f produt is 14 nd their sum is 5 g produt is 10 nd their sum is 3 h produt is 25 nd their sum is 0 i produt is 2 nd their sum is 1 j produt is 18 nd their sum is 7 2 Ftorise eh qudrti expression. x 2 þ 7x þ 12 b x 2 þ 12x þ 35 x 2 þ 5x þ 4 d x 2 þ 7x þ 10 e x 2 þ 9x þ 20 f t 2 þ 6t þ 5 g e 2 þ 5e þ 6 h h 2 þ 4h þ 4 i n 2 þ 11n þ 10 j 2 þ 11 þ 30 k d 2 þ 10d þ 24 l y 2 þ 15y þ 44 3 Ftorise eh qudrti expression. y 2 2y 3 b r 2 5r 14 h 2 3h 4 d w 2 7w 18 e e 2 6e 27 f Ftorise eh qudrti expression. x 2 þ 3x 4 b t 2 þ 5t 24 m 2 þ 2m 15 d 2 þ 2 e k 2 þ 5k 14 f w 2 þ 4w 12 g m 2 5m þ 4 h w 2 6m þ 8 i x 2 12x þ 35 j p 2 10p þ 24 k n 2 3n þ 2 l 2 6 þ 9 5 Ftorise eh expression given it is perfet squre. m 2 þ 4m þ 4 b p 2 þ 20p þ þ 25 6 Ftorise eh qudrti expression. Look for the highest ommon ftor first. 3m 2 þ 9m þ 6 b 2y 2 þ 2y 4 5t 2 10t 400 d 5e 4 þ 25e 3 120e 2 e x 3 x 2 110x f 4b 2 4b 168 g 4w 2 þ 4w 48 h i 2e 2 þ 18e þ 40 j 24 5t t 2 k 42 þ u u 2 l 28 þ 3x x 2 m 12 b b 2 n 7k 12 k 2 o 12x 35 x 2 100

36 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 7 Ftorise eh qudrti expression. h 2 3h 18 b 18 7t t 2 w 2 þ 8w þ 7 d k 2 12k 45 e v 2 þ 8v 20 f 3 2 þ 30 þ 75 g q 2 6q þ 5 h 2 4 þ 3 i 6x 2 þ 36x 96 j x 2 16x þ 64 k 8u 2 24u 32 l b 2 þ 11b þ 30 m y 2 12y þ 36 n 5r 2 5r 10 o 4l 2 8l 32 p g 2 24g þ 80 q 108 6d 2d 2 r 26 þ 11n n 2 Stge 5.3 Investigtion: Ftorising qudrti trinomils by grouping in pirs 1 We n ftorise the qudrti trinomil 6y 2 þ 19y þ 15 by rewriting it s 6y 2 þ 10y þ 9y þ 15, splitting up the middle term 19y. Ftorise 6y 2 þ 10y þ 9y þ 15 by grouping in pirs. b Ftorise 6y 2 þ 9y þ 10y þ 15 by grouping in pirs. Is your nswer the sme s tht for prt? In 6y 2 þ 19y þ 15, 6 is the oeffiient of y 2 while 15 is the onstnt term. Find the produt of 6 nd 15 nd the produt of 10 nd 9. Wht do you notie? 2 Ftorise 2 2 þ 13 þ 21 by rewriting it s 2 2 þ 6 þ 7 þ 21. b Ftorise 2 2 þ 13 þ 21 by rewriting it s 2 2 þ 7 þ 6 þ 21. Find the produt of 2 nd 21 nd the produt of 6 nd 7. Wht do you notie? 3-13 Ftorising qudrti expressions of the form x 2 þ bx þ We will now ftorise qudrti trinomils suh s 6x 2 þ 19x þ 15, where x 2 hs oeffiient tht is not ommon ftor. We do this by splitting the middle term into two terms nd then ftorise by grouping in pirs. Exmple 28 Ftorise 3x 2 þ 8x þ 4. We need to split up the middle term 8x. Find two numbers tht hve produt of 12 nd sum of 8. sum of 8 3x 2 + 8x + 4 = 3x 2 + 8x + 4 produt of 12 (3 4) 101

37 Chpter Produts nd ftors Stge 5.3 The two numbers re þ6 nd þ2, so we will split 8x into 6x nd 2x. ) 3x 2 þ 8x þ 4 ¼ 3x 2 þ 6x þ 2x þ 4 ¼ 3xðxþ 2Þþ2ðx þ 2Þ ¼ ðx þ 2Þð3x þ 2Þ Ftorising by grouping in pirs. Ftorising gin. Summry To ftorise qudrti trinomils of the form x 2 þ bx þ : find two numbers tht hve sum of b nd produt of use these two numbers to split the middle term bx into two terms ftorise by grouping in pirs Exmple 29 Ftorise eh qudrti expression. 5k 2 12k þ 4 b 9m 2 9m 4 6t 2 þ t 12 5k 2 12k þ ¼ 20. Find two numbers tht hve produt of 20 nd sum of 12. Sine the sum is negtive, one of the numbers must be negtive. Sine the produt is positive, both of the numbers must be negtive. They re 10 nd 2. Split 12k into 10k nd 2k. 5k 2 12k þ 4 ¼ 5k 2 10k 2k þ 4 ¼ 5kðk 2Þ 2ðk 2Þ Ftorising by grouping in pirs. ¼ ðk 2Þð5k 2Þ b 9m 2 9m ( 4) ¼ 36. Find two numbers with produt of 36 nd sum of 9. Sine the produt is negtive, one of the numbers must be negtive. They re 12 nd 3. 9m 2 9m 4 ¼ 9m 2 12m þ 3m 4 ¼ 3mð3m 4Þþ13m ð 4Þ ¼ ð3m 4Þð3m þ 1Þ 102

38 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 6t 2 þ t ( 12) ¼ 72. Find two numbers with produt of 72 nd sum of 1. They re 8 nd þ9. 6t 2 þ t 12 ¼ 6t 2 8t þ 9t 12 ¼ 2tð3t 4Þþ33t ð 4Þ ¼ ð3t 4Þð2t þ 3Þ Stge 5.3 Exmple 30 Ftorise eh qudrti expression b 10 7x 12x ¼ ¼ þ 3 4 ¼ 23 ½ ð3 4Þþ13 ð 4ÞŠ ¼ 23 ð 4Þð3 þ 1Þ b 10 7x 12x 2 ¼ 12x 2 7x þ 10 ¼ 12x 2 þ 7x 10 ¼ 12x 2 þ 15x 8x 10 ¼ 3x ½ ð4x þ 5Þ 24x ð þ 5ÞŠ ¼ 4x ð þ 5Þð3x 2Þ Tking out the HCF of 2 first. Produt ¼ 36, sum ¼ 9 Rerrnging the terms to mke the x 2 term first. Tking out ommon ftor of 1. Produt ¼ 120, sum ¼ 7 Exerise 3-13 Ftorising qudrti expressions of the form x 2 þ bx þ 1 Ftorise eh qudrti expression. 2x 2 þ 11x þ 5 b 4x 2 þ 13x þ 3 5x 2 þ 17x þ 6 d 6x 2 þ 19x þ 10 e 2w 2 þ 31w þ 15 f 4e 2 þ 15e þ 9 g 8f 2 þ 14f þ 3 h 3d 2 þ 5d þ 2 i 2b 2 þ 9b þ 7 j 5y 2 þ 16y þ 11 k 8g 2 þ 26g þ 15 l 6 2 þ 23 þ 21 2 Ftorise eh qudrti expression. 2y 2 11y þ 12 b 10k 2 19k þ 6 6e 2 13e þ 6 d 4b 2 13b þ 3 e 6w 2 23w þ 15 f 8t 2 þ 26t þ 15 g 9x 2 12x þ 4 h 12f 2 25f þ 12 i 4h 2 36h þ 81 j 5y 2 6y 11 k 4d 2 d 5 l 2m 2 3m 9 m 3t 2 t 30 n 6h 2 h 7 o 2y 2 5y 12 p q 15u 2 7u 4 r See Exmple 28 See Exmple

39 Chpter Produts nd ftors Stge 5.3 See Exmple 30 Worked solutions Ftorising qudrti expressions of the form x 2 þ bx þ MAT09NAWS10017 Worksheet Mixed ftoristions 3 Ftorise eh qudrti expression. 5m 2 þ 2m 7 b 6g 2 þ g 12 3p 2 þ 4p 4 d 7w 2 þ 6w 1 e 5y 2 þ 14y 3 f 3n 2 þ 10n 8 g 4b 2 þ 9b 9 h 8m 2 þ 10m 3 i 3x 2 þ 2x 16 4 Ftorise eh expression given it is perfet squre. 81w 2 180w þ 100 b 4y 2 þ 8y þ 4 25h 2 40h þ 16 5 Ftorise eh qudrti expression by first tking out ommon ftor. 6t 2 þ 10t 4 b 6g 2 þ 15g 36 24e 2 28e 12 d e 12t 2 þ 20t 8 f 25q 2 5q þ 6 g 12m 2 þ 14m 4 h 20 h 12h 2 i 18 þ 48 þ 24 2 j 15 þ 9z 6z 2 k 12d 2 þ 2d 30 l 22x 12 6x 2 6 Ftorise eh qudrti expression. 2 2 þ 5 þ 3 b 12m 2 32m þ 5 4x 2 þ 11x 3 d 7w 2 8w þ 1 e 4h 2 7h 15 f 8x 2 2x 3 g 5r 2 þ 26r þ 5 h 2d 2 15d þ 7 i 6n 2 7n 3 j 8 6m 9m 2 k l 15g 2 þ 19g þ 6 m 15 þ 14q 8q 2 n 3x 2 13x þ 14 o 16 8d 3d Mixed ftoristions MAT09NAWK00036 Puzzle sheet Ftorominoes MAT09NAPS10033 Puzzle sheet Produts nd ftors squresw MAT09NAPS00019 Summry Ftoristion strtegies Look for ny ommon ftors nd ftorise first If there re two terms, try ftorising using the differene of two squres If there re three terms, try ftorising s qudrti trinomil If there re four terms, try ftorising by grouping in pirs Algebri expression Tke out ny ommon ftors Two terms Three terms Four terms Ftorise if differene of two squres If qudrti trinomil, try to ftorise Try to ftorise by grouping in pirs 104

40 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 Exmple 31 Stge 5.3 Ftorise eh qudrti expression b 5 2 þ b 2 52b þ 24 d d 3 d 2 d þ ¼ ¼ 3ð þ 3Þð 3Þ b 5 2 þ 100 ¼ 5( 2 þ 20) 20b 2 52b þ 24 ¼ 45b 2 13b þ 6 ¼ 45b 2 10b 3b þ 6 ¼ 45bb ½ ð 2Þ 3ðb 2ÞŠ ¼ 4ðb 2Þð5b 3Þ d d 3 d 2 d þ 1 ¼ d 2 ðd 1Þ 1ðd 1Þ ¼ ðd 1Þ d 2 1 ¼ ðd 1Þðd þ 1Þðd 1Þ ¼ ðd 1Þ 2 ðd þ 1Þ Tking out the HCF of 3 first. Differene of two squres. Two terms but not differene of two squres. Ftorising by grouping in pirs. Differene of two squres. Exerise 3-14 Mixed ftoristions 1 Ftorise eh expression. m 2 16m þ 64 b 3d 2 3d 3d 2 4d 15 d 3k 15 5h þ hk e 25y 2 64 f 100f 2 64 g q 2 þ 3q 3pq h 3 þ 2g g 2 i 24b 2 þ 44b 40 j 25r 2 1 k b 3 þ b 2 þ b þ 1 l 4x 2 20x þ 25 m 4 d 5d 2 n b 3 b 2 b þ 1 o 8 2v 2 p mn 2 þ mnp þ 3mn þ 3mp q 2w 2 24w þ 72 r 36h 2 þ 12h þ 1 2 Ftorise eh expression. 15r 2 31rt 24t 2 b 4d 2 þ 4d þ 1 9g 2 36k 2 d e 3 3e 2 10e e 5(p þ q) 2 125(p q) 2 f 28x 2 7 g 2 b 2 þ 4 4b h þ 8 i 6 2 þ 13 5 j t 2 3t þ 5t 35 k 18p 2 þ 24p þ 8 l m 9x 2 27x þ 18x 54 n 2 2 b 6b 3 þ 9 o 2 2 þ 12 þ 18 p 25u 2 10u þ 1 q 4k 2 5k 21 r 48 3w 2 s 3 27s 2 t k 3 þ 4k 2 16k 64 u 5y 3 10y 2 þ 15y v m 3 n 4mn w x See Exmple

41 Chpter Produts nd ftors Stge 5.3 NSW 3-15 Ftorising lgebri frtions Exmple 32 Simplify eh expression. 10 þ 25b 5 b 9y2 16 6y þ 8 x 2 þ x 4x 4 d t2 3t þ 2 3t 2 5t 2 10 þ 25b 5 ¼ 5ð þ 5bÞ 5 ¼ þ 5b x 2 þ x 4x 4 ¼ xxþ ð 1Þ 4ðx þ 1Þ ¼ x ðx þ 1Þ 4 ðx þ 1Þ ¼ x 4 b 9y y þ 8 ¼ ð 3y þ 4 Þ ð 3y 4 Þ 23y ð þ 4Þ ð ¼ 3y þ 4 Þ ð 3y 4 Þ 2 ð3y þ 4Þ 3y 4 ¼ 2 d t 2 3t þ 2 3t 2 5t 2 ¼ ðt 2Þðt 1Þ ðt 2Þð3t þ 1Þ ¼ ðt 2Þðt 1Þ ðt 2Þð3t þ 1Þ ¼ t 1 3t þ 1 Exmple 33 Simplify eh expression. 4 x 2 þ x 2 x 2 1 b 3m m m 2 2m d2 þ 3d þ 2 d d2 þ d 3d þ 9 4 x 2 þ x 2 x 2 1 ¼ 4 xxþ ð 1Þ 2 ðx þ 1Þðx 1Þ 4ðx 1Þ ¼ xxþ ð 1Þðx 1Þ 2x xxþ ð 1Þðx 1Þ ¼ 4x 4 2x xxþ ð 1Þðx 1Þ ¼ 2x 4 xxþ ð 1Þðx 1Þ 2ðx 2Þ ¼ xxþ ð 1Þðx 1Þ Ftorising denomintors. Using ommon denomintors. 106

42 NEW CENTURY MATHS ADVANCED for the Austrlin Curriulum9 b 3m 6 3 8m 4 m 2 2m ¼ 3ðm 2Þ 3 8m 4 mm ð 2Þ ¼ 3 ðm 2Þ m m ðm 2Þ ¼ 6 d 2 þ 3d þ 2 d d2 þ d 3d þ 9 ¼ d2 þ 3d þ 2 d 2 3 3d þ 9 9 d 2 þ d ¼ ðd þ 2Þðd þ 1Þ ðd þ 3Þðd 3Þ 3 3ðd þ 3Þ ddþ ð 1Þ ¼ ðd þ 2Þðd þ 1Þ ðd þ 3Þðd 3Þ 3 3 ðd þ 3Þ d ðd þ 1Þ ¼ 3ðd þ 2Þ dd ð 3Þ Stge 5.3 Exerise 3-15 Ftorising lgebri frtions 1 Simplify eh expression. 3x þ 3y 3 b d y 1 1 y g ðk þ 5Þ k 2 25 j y 2 þ 9y þ 20 2y þ 10 m s2 þ 4s þ 4 s 2 s t 10r e w2 16 w þ 4 h þ 2 k k2 3k 4 k 2 16 n þ Simplify eh expression. 5 mmþ ð 1Þ þ 2 ðm þ 1Þðm þ 2Þ 3 ðb þ 2Þðb 1Þ þ 1 ðb 1Þðb 3Þ e 5 4h þ 4 þ 3 h 2 þ h 3 g r r þ 24 5 i k 2 3k 4 k k 2 1 b d f h j f i l o b 2 5d 5t d 2 t 2 m n þ m n m 2 n þ 5 2 p þ 4p 2 8 2p ðw þ 5Þðw þ 3Þ 4 wwþ ð 3Þ 2 k 2 þ k 3 k d 2 þ 3d þ 2 4 d þ 2 3 d 2 þ 2d þ d d q 2 1 þ 3 q þ 1 See Exmple 32 See Exmple

43 Chpter Produts nd ftors Stge Simplify eh expression. 3m þ 9 2 d 3k þ 6 5 g r þ t j 3 4m m þ k k þ 2 t 2 r 2 3 r2 rt 5r þ 5t y þ 2 7y þ y 15y m d2 þ d d þ 3 4 6d d 2 9 b e h k n 5d 10 3d 9 3 5d 15 8d 16 5h 3 6h þ 18 3h þ 9 h 2 þ h 20m þ 16 7m 7 3 7m 5m þ 4 5 x x þ 4 1 f 2 6f þ f 2 9 f i l o 4 e þ 2 3 e2 þ 2e 8e þ 3b b2 8 p 2 þ 2p þ 1 4p 4 p p 2 þ p 4n þ 8 4 6n þ 12 n þ 5 5n þ 25 3f þ 6 f 2 þ f 6 4 f 2 2f 8 f 2 f 12 Power plus 1 For this omposite shpe, write n expression for: A the length of CD b the length of BC 3u + 2t its perimeter d its re F u + t C B 2u + 5t 2 Find the verge of 6r, 2r þ 8, r 5, 2r, r þ 7 nd 3r þ 8. 3 A retngulr grden hs its longer sides eh 5 m longer thn its shorter sides. If its longer sides eh hve length y m, then write simplified expression for: the re of the grden b the perimeter of the grden 4 How mny hours does it tke r to trvel distne of 20t 2 kilometres t n verge speed of 4t km/h? 5 Expnd nd simplify eh expression. ( þ b)( þ b þ ) b (x 2)(x þ y 3) (p þ 3)(p 4)(p 5) d (x 1)(y 1)(z 1) e (wt 3r) 2 f (x 3 þ 2) 2 6 Ftorise eh expression. n 2 þ 4mn þ 4m 2 b x 2 2xy þ y 2 25x 2 40xy þ 16y 2 d b þ 45b 2 e 4 þ 2 2 þ 1 f 8 7t 2 t 4 g x 4 1 h ( þ b) 2 2 i (p þ q) 2 (p q) 2 7 If x 2 þ bx þ ¼ (x þ p)(x þ q), wht re the signs of p nd q if: b nd re both positive? b b nd re both negtive? b is positive nd is negtive? d b is negtive nd is positive? D u t E 108